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ISC Class 12 Probability & Statistics — practice questions

Explore 50 free Probability & Statistics questions in ISC Class 12 Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Probability & Statistics questions and exercises with answers

Showing questions 1–30 of 50.

Simplify: arcsin(sin(x)) for x ∈ [π/2, π]:Read the answer and worked solution →Solve 2 arctan(x) = π/3:Read the answer and worked solution →Range of f(x) = arcsin(x) + arccos(x):Read the answer and worked solution →d/dx [arctan(x²)] =Read the answer and worked solution →arctan(x) - arctan(y) = (for xy > -1):Read the answer and worked solution →cos(2 arctan(x)) =Read the answer and worked solution →d/dx arcsin(sqrt(x)) =Read the answer and worked solution →Number of solutions to arctan(x) = π/3 in real x:Read the answer and worked solution →$\int \dfrac{1}{x}\,dx$ equalsRead the answer and worked solution →$\int a^{x}\,dx$ (with $a>0,\ a\neq 1$) equalsRead the answer and worked solution →$\int \sec^2 x\,dx$ equalsRead the answer and worked solution →$\int \dfrac{x^3-1}{x^2}\,dx$ equalsRead the answer and worked solution →$\int \operatorname{cosec} x\,(\operatorname{cosec} x+\cot x)\,dx$ equalsRead the answer and worked solution →$\int \dfrac{1-\sin x}{\cos^2 x}\,dx$ equalsRead the answer and worked solution →If $F$ is the antiderivative of $f(x)=4x^3-6$ with $F(0)=3$, then $F(x)$ isRead the answer and worked solution →The antiderivative of $\left(\sqrt{x}+\dfrac{1}{\sqrt{x}}\right)$ equalsRead the answer and worked solution →Using inspection, an antiderivative of $\cos 2x$ isRead the answer and worked solution →$\int 2x\,\sin(x^2+1)\,dx$ equalsRead the answer and worked solution →$\int \sin mx\,dx$ equalsRead the answer and worked solution →$\int \tan x\,dx$ equalsRead the answer and worked solution →$\int \cot x\,dx$ equalsRead the answer and worked solution →$\int \sec x\,dx$ equalsRead the answer and worked solution →$\int \dfrac{2x}{1+x^2}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{(\log x)^2}{x}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{e^{\tan^{-1}x}}{1+x^2}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{10x^9+10^{x}\log_e 10}{x^{10}+10^{x}}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{dx}{\sin^2 x\cos^2 x}$ equalsRead the answer and worked solution →$\int \cos^2 x\,dx$ equalsRead the answer and worked solution →$\int \sin^3 x\,dx$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x^2+a^2}$ equalsRead the answer and worked solution →

Worked practice questions

Simplify: arcsin(sin(x)) for x ∈ [π/2, π]:

  1. -x
  2. π - x (since sin(π-x) = sin x and π - x ∈ [0, π/2])
  3. x
  4. π + x
Show answer and solution

Correct answer: B. π - x (since sin(π-x) = sin x and π - x ∈ [0, π/2])

For x ∈ [π/2, π]: sin x = sin(π - x), and π - x ∈ [0, π/2] which IS in arcsin's principal range. **So arcsin(sin x) = π - x.**

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Solve 2 arctan(x) = π/3:

  1. x = 0
  2. x = tan(π/6) = 1/sqrt(3)
  3. x = 1
  4. x = sqrt(3)
Show answer and solution

Correct answer: B. x = tan(π/6) = 1/sqrt(3)

arctan(x) = π/6 → x = tan(π/6) = 1/√3 ≈ 0.577.

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Range of f(x) = arcsin(x) + arccos(x):

  1. Empty
  2. [0, π]
  3. {π/2} (constant function, equals π/2 always on [-1, 1])
  4. [-π/2, π/2]
Show answer and solution

Correct answer: C. {π/2} (constant function, equals π/2 always on [-1, 1])

For all x ∈ [-1, 1]: arcsin(x) + arccos(x) = π/2 (constant identity). **Hence range is single point {π/2}.**

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